Uncle Petros and Goldbach's Conjecture

by Apostolos Doxiadis

  ‘I know the whole truth, Uncle Petros,’ I continued fervently. ‘You got to within a hair’s breadth of the proof! You were almost there … Almost … All but the final step …’ — my voice was coming out in a humming, deep chant—’ … and then, you lost your nerve! You chickened out, Uncle dearest, didn’t you? What happened! Did you run out of willpower or were you just too scared to follow the path to its ultimate conclusion? Whatever the case, you’d always known it deep inside: the fault is not with the Incompleteness of Mathematics!’

  My last words had made him recoil and I thought I might as well play the part to the hilt: I grabbed him by the shoulders and shouted straight into his face. ‘Face it, Uncle! You owe it to yourself, can’t you see that? To your courage, to your brilliance, to all those long, fruitless, lonely years! The blame for not proving Goldbach’s Conjecture is all your own — just as the triumph would have been totally yours if you’d succeeded! But you didn’t succeed! Goldbach’s Conjecture is provable and you knew that all along! It’s just that you didn’t manage to prove it! You failed — you failed, God damn it, and you’ve got to admit it, at last!’

  I had run out of breath.

  As for Uncle Petros, for a slight moment his eyes closed and he wavered. I thought that he was going to pass out, but no — he instantly came to, his inner turmoil now unexpectedly melting into a soft, mellow smile.

  I smiled too: naively, I thought that my wild ranting had miraculously achieved its purpose. In fact, at that moment I would have made a bet that his next words would be something like: ‘You are absolutely right. I failed. I admit it. Thank you for helping me do it, most favoured of nephews. Now, I can die happy.’

  Alas, what he actually said was: ‘Will you be a good boy and go get me five more kilos of beans?’

  I was stunned — all of a sudden he was the ghost and I Hamlet.

  ‘We — we must finish our discussion first,’ I faltered, too shocked for anything stronger.

  But then he started pleading: ‘Please! Please, please, please get me some more beans!’

  His tone was so intolerably pathetic that my defences crumbled to dust. For better or for worse, I knew that my experiment in enforced self-confrontation had ended.

  Buying uncooked beans in a country where people don’t do their grocery shopping in the middle of the night was a worthy challenge to my developing entrepreneurial skills. I drove from taverna to taverna, beguiling the cooks into selling me from their pantry stock a kilo here, half a kilo there, until I accumulated the required quantity. (It was probably the most expensive five kilos of beans ever.)

  When I got back to Ekali, it was past midnight. I found Uncle Petros waiting for me at the garden gate.

  ‘You are late!’ was his only greeting.

  I could see that he was in a state of tremendous agitation.

  ‘Everything all right, Uncle?’

  ‘Are these the beans?’

  ‘They are, but what’s the matter? What are you so worked up about?’

  Without answering he grabbed the bag. ‘Thank you,’ he said and began to close the gate.

  ‘Shan’t I come in?’ I asked, surprised.

  ‘It’s too late,’ he said.

  I was reluctant to leave him until I found out what was going on.

  ‘We don’t have to talk mathematics,’ I said. ‘We can have a little game of chess or, even better, drink some herbal tea and gossip about the family.’

  ‘No,’ he said with finality. ‘Goodnight.’ He walked fast towards his small house.

  ‘When is the next lesson?’ I shouted after him.

  ‘I’ll call you,’ he said, went in and banged the door behind him.

  I remained standing on the pavement for a while, wondering what to do, whether to attempt once again to enter the house, to talk to him, to see if he was all right. But I knew he could be stubborn as a mule. Anyway, our lesson and my nocturnal search for beans had drained me of all energy

  Driving back to Athens I was pestered by my conscience. For the first time, I questioned my course of action. Could my high-handed stance, supposedly intended to lead Uncle Petros into a therapeutic showdown, have been nothing more than my own need to get even, an attempt to avenge the trauma he’d inflicted on my teenage self? And, even if that weren’t so, what right did I have to make the poor old man face the phantoms of his past, despite himself? Had I seriously considered the consequences of my inexcusable immaturity? The unanswered questions abounded, but still, by the time I got home I had rationalized myself out of the moral tight spot: the distress I’d obviously caused Uncle Petros had most probably been the necessary — the obligatory — step in the process of his redemption. What I’d told him was, after all, too much to digest at one go. Obviously the poor man only needed a chance to think things over in peace. He had to admit his failure to himself, before he could do so to me…

  But if that was the case, why the extra five kilos of beans?

  A hypothesis had begun to form in my mind, but it was too outrageous to be given serious consideration — until morning anyway.

  Nothing in this world is truly new — certainly not the high dramas of the human spirit. Even when one such appears to be an original, on closer examination you realize it’s been enacted before, with different protagonists, of course, and quite possibly with many variations in its development. But the main argument, the basic premise, repeats the same old story.

  The drama played out during Petros Papachristos’ final days is the last in a triad of episodes from the history of mathematics, unified by a single theme: the Mystery-solution to a Famous Problem by an Important Mathematician.*

  By majority consent, the three most famous unsolved mathematical problems are: (a) Fermat’s Last Theorem, (b) the Riemann Hypothesis and (c) Goldbach’s Conjecture.

  In the case of Fermat’s Last Theorem, the mystery-solution existed from its first statement: in 1637, while he was studying Diophantus’ Arithmetica, Pierre de Fermat made a note in the margin of his personal copy, right next to proposition II.8 referring to the Pythagorean theorem, in the form x2 + y2 = z2. He wrote: ‘It is impossible to separate a cube into two cubes, or a biquadrate (fourth power) into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvellous proof of this, which, however, this margin is not large enough to contain.’

  After the death of Fermat his son collected and published his notes. A thorough search of his papers, however, failed to reveal the demonstratio mirabilis, the ‘marvellous proof’ that his father claimed to have found. Equally in vain have mathematicians ever since sought to rediscover it.* As for the verdict of history on the existence of the mystery-solution: it’s ambiguous. Most mathematicians today doubt that Fermat indeed had a proof. The worst-case theory has it that he was consciously lying, that he had not verified his guess and his margin-note was mere bragging. What’s likelier, however, is that he was mistaken, the demonstratio mirabilis crippled by an undetected fault.

  In the case of the Riemann Hypothesis, the mystery-solution was in fact a metaphysical practical joke, with G. H. Hardy as its perpetrator. This is how it happened:

  Preparing to board a cross-Channel ferry during a bad storm, the confirmed atheist Hardy sent off to a colleague a postcard with the message: ‘I have the proof to the Riemann Hypothesis.’ His reasoning was that the Almighty, whose sworn enemy he was, would not permit him to reap such an exalted undeserved reward and would therefore see to his safe arrival, in order to have the falsity of his claim exposed.

  The mystery-solution of Goldbach’s Conjecture completes the triad.

  On the morning after our last lesson, I telephoned Uncle Petros. At my insistence, he had recently agreed to have a line installed, on the condition that only I, and no one else, would know the number.

  He answered sounding tense and distant. ‘What do you want?’

  ‘Oh, I just called to say hello,’ I said. ‘Also to apologize. I
think I was unnecessarily rude last night.’

  There was a pause.

  ‘Well,’ he said, ‘actually I’m busy at the moment. Why don’t we talk again…shall we say next week?’

  I wanted to assume that his coldness was due to the fact that he was upset with me (as he had every reason to be, after all) and merely expressing his resentment. Still, I felt a nagging unease.

  ‘Busy with what, Uncle?’ I persisted.

  Another pause.

  ‘I-I’ll tell you about it some other time.’

  He was obviously eager to hang up so, before he could cut me off, I impulsively blurted out the suspicion that had taken shape during the night.

  ‘You wouldn’t by any chance have resumed your researches, would you, Uncle Petros?’

  I heard a sharp intake of breath. ‘Who — who told you that?’ he said hoarsely.

  I tried to sound casual. ‘Oh, come on, give me some credit for having come to know you. As if it needed telling!’

  I heard the click of his hanging up. My God — I was right! The crazy old fool had gone off his rocker. He was trying to prove Goldbach’s Conjecture!

  My guilty conscience stung me. What had I done? Humankind indeed cannot stand very much reality — Sammy’s theory of Kurt Gödel’s insanity also applied, in a different way, to Uncle Petros. I had obviously pushed the poor old man to his uttermost limit and then beyond it. I’d aimed straight at his Achilles heel and hit it. My ridiculous simple-minded scheme to force him into self-confrontation had destroyed his fragile defences. Heedlessly, irresponsibly, I had robbed him of the carefully nurtured justification of his failure: the Incompleteness Theorem. But I had put nothing in its place to sustain his shattered self-image. As his extreme reaction now showed, the exposure of his failure (to himself, more than to me) had been more than he could bear. Stripped of his cherished excuse he had taken, of necessity, the only way left for him to go: madness. For what else was the endeavour to search, in his late seventies, for the proof that he had failed to find when he was at the peak of his powers? If that wasn’t total irrationality, what was?

  I walked into my father’s office filled with apprehension. Much as I hated to allow him into the charmed circle of my bond with Uncle Petros, I felt obliged to let him know what had happened. He was after all his brother, and any suspicion of serious illness was certainly a family matter. My father dismissed my self-recriminations about causing the crisis as so much poppycock. According to the official Papachristos world-view, a man had only himself to blame for his psychological condition, the only acceptable external reason for emotional discomfort being a serious drop in the price of stocks. As far as he was concerned, his older brother’s behaviour had always been bizarre, and one more instance of eccentricity was definitely not to be taken seriously

  ‘In fact,’ he said, ‘the condition you describe — absent-mindedness, self-absorption, abrupt changes of mood, irrational demands for beans in the middle of the night, nervous tics, etc. — reminds me of how he was carrying on when we visited him in Munich, back in the late twenties. Then, too, he was behaving like a madman. We’d be at a nice restaurant enjoying our wurst and he’d be squirming around as if there were nails in his chair, his face twitching like mad.’

  ‘Quod erat demonstrandum,’ I said. That’s exactly it. He’s back doing mathematics. In fact, he’s back working on Goldbach’s Conjecture — ridiculous as that may sound at his age.’

  My father shrugged. ‘It’s ridiculous at any age,’ he said. ‘But why worry? Goldbach’s Conjecture has already done him all the harm possible. Nothing worse can come of it.’

  But I wasn’t so sure about that. In fact, I was quite certain that a lot worse things could be in store for us. Goldbach’s resurrection was bound to stir up unfulfilled passions, to aggravate deep-buried, terrible, unhealed wounds. His absurd new application to the old problem boded no good.

  After work that evening, I drove to Ekali. The ancient VW beetle was parked outside the house. I crossed the front yard and rang the bell. There was no response, so I shouted: ‘Open up, Uncle Petros; it’s me!’

  For a few moments I feared the worst, but then he appeared at a window and stared vaguely in my direction. There was no sign of his usual pleasure at seeing me, no surprise, no greeting — he just stared.

  ‘Good afternoon,’ I said. ‘I just came by to say hello.’

  His normally serene face, the face of a stranger to life’s usual worries, was now marked by extreme tension, his skin pale, his eyes red with sleeplessness, his brow furrowed with concern. He was also unshaven, the first time I’d seen him so. His stare continued absent, unfocused. I wasn’t even sure he knew who I was.

  ‘Come on, Uncle dear, please open up for the most favoured,’ I said with a fatuous smile.

  He disappeared and after a while the door creaked open. He stood there, blocking my entry, wearing his pyjama bottoms and a wrinkled vest. It was evident he didn’t want me to enter.

  ‘What’s wrong, Uncle?’ I asked. ‘I’m worried about you.’

  ‘Why should you be worried?’ he said, now forcing himself to sound normal. ‘Everything’s fine.’

  ‘Are you sure?’

  ‘Of course I’m sure.’

  Then, with a snappy gesture, he beckoned me closer. After quickly, anxiously glancing around, he leaned towards me, his lips almost touching my ear, and whispered: ‘I saw them again.’

  I didn’t understand. ‘Who did you see?’

  ‘The girls! The twins, the number 2100!’

  I remembered the strange apparitions of his dreams.

  ‘Well,’ I said, trying to sound as casual as possible. ‘If you are once again involved with mathematical research, you are once again having mathematical dreams. Nothing strange about that…’

  I wanted to keep him talking so as to (figuratively, but if need be also literally) put a foot in the door. I had to get some sense of how bad his condition was.

  ‘So what happened, Uncle,’ I asked, feigning great interest in the matter. ‘Did the girls speak to you?’

  ‘Yes,’ he said, ‘they gave me a … ‘His voice quickly trailed off, as if he was afraid he’d said too much.

  ‘A what?’ I asked. ‘A clue?’

  He became suspicious again. ‘You mustn’t tell,’ he said sternly.

  ‘Mum’s the word,’ I said.

  He had started to close the door. Convinced now that his situation was extremely serious and that the time had come for emergency action, I grasped the knob and started to push. As he felt my force, he tensed up, gritted his teeth and struggled to prevent me from entering, his face contorted to a grimace of desperation. Fearing the effort might be too much for him (he was nearing eighty, after all) I reduced the pressure a bit for a final attempt at reason.

  Of all the possible stupid things I could have said to him, I chose this: ‘Remember Kurt Gödel, Uncle Pet-ros! Remember the Incompleteness Theorem — Goldbach’s Conjecture is unprovable!’

  Instantly, his expression changed from despair to wrath. ‘Fuck Kurt Gödel,’ he barked, ‘and fuck his Incompleteness Theorem!’ With an unexpected upsurge of strength, he overcame my resistance and slammed the door shut in my face.

  I rang the bell again and again, banged the door with my fist and shouted. I tried threats, reasoning and pleading, but nothing worked. When a torrential October rain began to fall I hoped that, mad or not, Uncle Petros might be moved by mercy and let me in. But he wasn’t. I left, soaking wet and very worried.

  From Ekali I drove straight to our family doctor and explained the situation. Without altogether ruling out serious mental disturbance (possibly triggered by my unwarranted interference in his defence mechanisms) he suggested two or three organic problems as likelier causes of my uncle’s transformation. We decided to go to his house first thing the next morning, force our way in if necessary, and submit him to a thorough medical examination.

  That night I couldn’t sleep. The rain was getting strong
er, it was past two o’clock and I was sitting at home hunched in front of the chessboard, just as Uncle Petros must have been on innumerable sleepless nights, studying a game from the recent world championship. Yet my concern kept interfering and I couldn’t concentrate.

  When I heard the ringing I knew it was he, even though he’d never yet initiated a call on his newly installed telephone.

  I jumped up and answered.

  ‘Is that you, Nephew?’ He was obviously all worked up about something.

  ‘Of course it’s me, Uncle. What’s wrong?’

  ‘You must send me someone. Now!’

  I was alarmed. ‘“ Someone”? A doctor you mean?’

  ‘What use would a doctor be? A mathematician, of course!’

  I humoured him: ‘I’m a mathematician, Uncle; I’ll come right away! Just promise to open the door, so I won’t catch pneumonia and —’

  He obviously didn’t have time for irrelevancies. ‘Oh hell!’ he grunted and then: ‘All right, all right, you come, but bring another one as well!’

  ‘Another mathematician?’

  ‘Yes! I must have two witnesses! Hurry!’

  ‘But why do the witnesses have to be mathematicians?’

  Naively, I had thought at first he wanted to write his will.

  ‘To understand my proof!’

  ‘Proof of what?’

  ‘Goldbach’s Conjecture, you idiot — what else!’

  I chose my next words very carefully. ‘Look, Uncle Petros,’ I said, ‘I promise to be with you as soon as my car will get me there. Let’s be reasonable, mathematicians aren’t kept on call — how on earth can I get one at two o’clock in the morning? You’ll tell me all about your proof tonight and tomorrow we will go together —’

  But he cut me off, screaming. ‘No, no, no! There’s no time for any of that! I need my two witnesses and I need them now!’ Then he broke down and started sobbing. ‘O nephew, it’s so … it’s so …’

  ‘So what, Uncle? Tell me!’

  ‘Oh, it’s so simple, so simple, my dearest boy! How is it possible that all those years, those endless years, I hadn’t realized how blessedly simple it was!’